Pricing and greening strategies in a dual-channel supply chain with cost and profit sharing contracts

Abstract

The expanding customer consciousness of ecological sustainability has motivated supply chain members to participate in green activities. In this paper, the coordination issue of a dual-channel supply chain is studied under consideration of the greening level of the items. The two-stage supply chain consists of a manufacturer and a retailer. The manufacturer is responsible for keeping the item’s greening level and sells the products through two channels (a) a direct online channel and (b) a traditional retail channel. Market demand depends on the selling price and greening level of the item. Furthermore, the pricing and greening strategies of the channel members are discussed under the centralized and decentralized scenarios. Compared to the centralized scenario, optimum pricing at the retail channel is higher in the decentralized scenario while the greening level of products is low. The outcomes exhibit that the profit of the supply chain in a decentralized scenario decreases compared to the centralized scenario. To enhance the supply chain profit, we have developed two coordinate mechanisms of the decentralized scenario with a cost-sharing contract and a profit-sharing contract. Our analysis shows that the profit-sharing contract can realize the coordination, but the cost-sharing contract cannot. A numerical example has been demonstrated to quantify the effectiveness of different contracts, and the model’s finding is demonstrated.

Appendix

1.1 Appendix A

In the centralized case, the $1^{st}$ order derivative of $\Pi _{sc}^C$ from (6) w.r.to $p_r$, $p_o$ and g are

$$\begin{aligned}&\frac{\partial \Pi _{sc}^C }{\partial p_r}= \gamma _1 g - 2b_1p_r +b_1c + b_1^{'} p_o + b_2^{'} p_o - b_2^{'} c +a \alpha \end{aligned}$$

(40)

$$\begin{aligned}&\frac{\partial \Pi _{sc}^C }{\partial p_o}= \gamma _2 g - 2b_2p_o +b_2c + b_2^{'} p_r + b_1^{'} p_r - b_1^{'} c +a (1-\alpha ) \end{aligned}$$

(41)

$$\begin{aligned}&\frac{\partial \Pi _{sc}^C }{\partial g} = -e g + \gamma _1 (p_r-c) +\gamma _2 (p_o-c) \end{aligned}$$

(42)

To prove the optimality of the solution, equation (7), (8) and (9); the corresponding hessian matrix is calculated as follows:

$$\begin{aligned} H(\Pi _{sc}^C) = \begin{bmatrix} \frac{\partial ^2 \Pi _{sc}^c }{\partial p_r^2} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial p_r \partial p_o} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial p_r \partial g} \\ \frac{\partial ^2 \Pi _{sc}^C }{\partial p_o \partial p_r} &{} \frac{\partial ^2 \Pi _{sc}^C}{\partial p_o^2} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial p_o \partial g} \\ \frac{\partial ^2 \Pi _{sc}^C }{\partial g \partial p_r} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial g \partial p_o} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial g^2} \end{bmatrix} = \begin{bmatrix} &{} -2b_1 &{} b_1^{'}+b_2^{'} &{} \gamma _1 \\ &{} b_1^{'}+b_2^{'} &{} -2b_2 &{} \gamma _2 \\ &{} \gamma _1 &{} \gamma _2 &{} -e \end{bmatrix} \end{aligned}$$

The above Hessian matrix ($H(\Pi _{sc}^C)$) is must be negative definite in nature and the principal minors ($H^{i\times i}$, $i=1,2$) should have the following conditions

$|(H_{1*1})|$ = $\frac{\partial ^2 \Pi _{sc}^C }{\partial p_r^2} = -2b_1 <0$,

$|(H_{2*2})|$ = $\det \begin{bmatrix} \frac{\partial ^2 \Pi _{sc}^C }{\partial p_r^2} &{} \frac{\partial ^2 \Pi _{sc}^C }{\partial p_r \partial p_o} \\ \frac{\partial ^2 \Pi _{sc}^C }{\partial p_o \partial p_r} &{} \frac{\partial ^2 \Pi _{sc}^C}{\partial p_o^2} \end{bmatrix}$ = $\det \begin{bmatrix} &{} -2b_1 &{} b_1^{'}+b_2^{'} \\ &{} b_1^{'}+b_2^{'} &{} -2b_2 \end{bmatrix}$ $>0$ .

Therefore, $|(H_{2*2})|=4b_1b_2-(b_1^{'}+b_2^{'})^2 >0$ if $\frac{4b_1b_2}{(b_1^{'}+b_2^{'})^2} > 1$ holds

$|H(\Pi _{sc}^C)|=-4e b_1 b_2 + 2 b_1 \gamma _2^2 + 2b_2 \gamma _1^2 +2 \gamma _1 \gamma _2 (b_1^{'}+b_2^{'})+e(b_1^{'}+b_2^{'})^2 <0$ if $\frac{4e b_1 b_2}{2 b_1 \gamma _2^2 + 2b_2 \gamma _1^2 +2 \gamma _1 \gamma _2 (b_1^{'}+b_2^{'})+e(b_1^{'}+b_2^{'})^2}>1$ holds.

1.2 Appendix B

The second order derivative of $\Pi _{m}^{DC}$ with respect to $p_o$, w and g are,

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{DC} }{\partial p_o^2}=-2 \big (b_2- b_1^{'}K\big )<0 \quad \text {if} \quad b_2> b_1^{'}K \quad \text {holds.} \end{aligned}$$

(43)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{DC} }{\partial w^2}= -b_1 <0 \end{aligned}$$

(44)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{DC} }{\partial g^2}= -e<0 \end{aligned}$$

(45)

The corresponding Hessian matrix is as follows,

$$\begin{aligned} H(\Pi _{m}^{DC}) = \begin{bmatrix} \frac{\partial ^2 \Pi _{m}^{DC}}{\partial p_o^2} &{} \frac{\partial ^2 \Pi _{m}^{DC} }{\partial p_o \partial w} &{} \frac{\partial ^2 \Pi _{m}^{DC} }{\partial p_o \partial g} \\ \frac{\partial ^2 \Pi _{m}^{DC} }{\partial w \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{DC}}{\partial w^2} &{} \frac{\partial ^2 \Pi _{m}^{DC} }{\partial w \partial g} \\ \frac{\partial ^2 \Pi _{m}^{DC}}{\partial g \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{DC} }{\partial g \partial w} &{} \frac{\partial ^2 \Pi _{m}^{DC} }{\partial g^2} \end{bmatrix} = \begin{bmatrix} &{} -2 \big (b_2- b_1^{'}K\big ) &{} \frac{b_1^{'}}{2}+b_1K &{} \gamma _1 K+\gamma _2 \\ &{} \frac{b_1^{'}}{2}+b_1K &{} -b_1 &{} \frac{\gamma _1}{2} \\ &{} \gamma _1 K+\gamma _2 &{} \frac{\gamma _1}{2} &{} -e \end{bmatrix} \end{aligned}$$

The corresponding principal minor

det$(H_{2*2}) > 0$ gives $2b_1 \big (b_2- b_1^{'}K\big ) > \big (\frac{b_1^{'}}{2}+b_1K \big )^2$.

and

det$(H_{3*3}) < 0$ gives $e \big (\frac{b_1^{'}}{2}+b_1K \big )^2 +\gamma _1 \big (\gamma _1 K+\gamma _2 \big ) \big (\frac{b_1^{'}}{2}+b_1K \big ) +b_1 \big (\gamma _1 K+\gamma _2 \big )^2 < 2 \big (b_2- b_1^{'}K\big ) \big (b_1 e - \frac{\gamma _1^2}{4}\big )$.

1.3 Appendix C

The second order derivative of $\Pi _{m}^{CS}$ with respect to $p_o$, w and g are,

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{CS} }{\partial p_o^2}=-2 \big (b_2- b_1^{'}K\big )<0 \quad \text {if} \quad b_2> b_1^{'}K \quad \text {holds.} \end{aligned}$$

(46)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{CS} }{\partial w^2}= -b_1 <0 \end{aligned}$$

(47)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{CS} }{\partial g^2}= -e<0 \end{aligned}$$

(48)

The corresponding Hessian matrix is as follows,

$$\begin{aligned} H(\Pi _{m}^{CS}) = \begin{bmatrix} \frac{\partial ^2 \Pi _{m}^{CS}}{\partial p_r^2} &{} \frac{\partial ^2 \Pi _{m}^{CS} }{\partial p_r \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{CS} }{\partial p_r \partial g} \\ \frac{\partial ^2 \Pi _{m}^{CS} }{\partial p_o \partial p_r} &{} \frac{\partial ^2 \Pi _{m}^{CS}}{\partial p_o^2} &{} \frac{\partial ^2 \Pi _{m}^{CS} }{\partial p_o \partial g} \\ \frac{\partial ^2 \Pi _{m}^{CS}}{\partial g \partial p_r} &{} \frac{\partial ^2 \Pi _{m}^{CS} }{\partial g \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{CS} }{\partial g^2} \end{bmatrix} = \begin{bmatrix} &{} -2 \big (b_2- b_1^{'}K\big ) &{} \frac{b_1^{'}}{2}+b_1K &{} \gamma _1 K+\gamma _2 \\ &{} \frac{b_1^{'}}{2}+b_1K &{} -b_1 &{} \frac{\gamma _1}{2} \\ &{} \gamma _1 K+\gamma _2 &{} \frac{\gamma _1}{2} &{} -\delta e \end{bmatrix} \end{aligned}$$

The corresponding principal minor

det$(H_{2*2}) > 0$ gives $2b_1 \big (b_2- b_1^{'}K\big ) > \big (\frac{b_1^{'}}{2}+b_1K \big )^2$.

and

det$(H_{3*3}) < 0$ gives $\delta e \big (\frac{b_1^{'}}{2}+b_1K \big )^2 +\gamma _1 \big (\gamma _1 K+\gamma _2 \big ) \big (\frac{b_1^{'}}{2}+b_1K \big ) +b_1 \big (\gamma _1 K+\gamma _2 \big )^2 < 2 \big (b_2- b_1^{'}K\big ) \big (b_1 \delta e - \frac{\gamma _1^2}{4}\big )$.

1.4 Appendix D

The second order derivative of $\Pi _{m}^{PS}$ with respect to $p_o$, w and g are,

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{PS} }{\partial p_o^2}=-2 \big (b_2- b_1^{'}K\big )<0 \quad \text {if} \quad b_2> b_1^{'}K \quad \text {holds.} \end{aligned}$$

(49)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{PS} }{\partial w^2}= -b_1 <0 \end{aligned}$$

(50)

$$\begin{aligned}&\frac{\partial ^2 \Pi _{m}^{PS} }{\partial g^2}= -e<0 \end{aligned}$$

(51)

The corresponding Hessian matrix is as follows,

$$\begin{aligned} H(\Pi _{m}^{PS}) = \begin{bmatrix} \frac{\partial ^2 \Pi _{m}^{PS}}{\partial p_r^2} &{} \frac{\partial ^2 \Pi _{m}^{PS} }{\partial p_r \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{PS} }{\partial p_r \partial g} \\ \frac{\partial ^2 \Pi _{m}^{PS} }{\partial p_o \partial p_r} &{} \frac{\partial ^2 \Pi _{m}^{PS}}{\partial p_o^2} &{} \frac{\partial ^2 \Pi _{m}^{PS} }{\partial p_o \partial g} \\ \frac{\partial ^2 \Pi _{m}^{PS}}{\partial g \partial p_r} &{} \frac{\partial ^2 \Pi _{m}^{PS} }{\partial g \partial p_o} &{} \frac{\partial ^2 \Pi _{m}^{PS} }{\partial g^2} \end{bmatrix} = \begin{bmatrix} &{} K_1 {b_1^{'}}^2 - 2 (b_2-b_1^{'}K) &{} \frac{(1-\sigma )b_1^{'}}{2}+b_1K &{} K_1b_1^{'} \gamma _1 +\gamma _1 K+\gamma _2 \\ &{} \frac{(1-\sigma )b_1^{'}}{2}+b_1K &{} -b_1 (1-\sigma +K_1 b_1) &{} \frac{(1-\sigma )\gamma _1}{2} \\ &{} K_1b_1^{'} \gamma _1 +\gamma _1 K+\gamma _2 &{} \frac{(1-\sigma )\gamma _1}{2} &{} K_1 \gamma _1^2-e \end{bmatrix} \end{aligned}$$

The corresponding principal minor

det$(H_{2*2}) > 0$ gives $b_1 \big (2b_2-2Kb_1^{'}-K_1{b_1^{'}}^2 \big ) \big (1-\sigma +K_1 b_1 \big ) > \big \{ \frac{(1-\sigma )b_1^{'}}{2}+b_1K \big \}^2$.

and

det$(H_{3*3}) < 0$ gives $\big (2b_2-2Kb_1^{'}-K_1{b_1^{'}}^2 \big ) \Big \{ b_1(1-\sigma +K_1 b_1) \big (K_1 \gamma _1^2-e \big ) + \Big (\frac{(1-\sigma )\gamma _1}{2} \Big )^2 \Big \} + \big (K_1b_1^{'} \gamma _1 +\gamma _1 K+\gamma _2 \big ) \Big \{ 2 \big (\frac{(1-\sigma )b_1^{'}}{2}+b_1K \big ) \big (\frac{(1-\sigma )\gamma _1}{2} \big ) +b_1 (1-\sigma +b_1K_1) \big (K_1b_1^{'} \gamma _1 +\gamma _1 K+\gamma _2 \big ) \Big \} < \big (\frac{(1-\sigma )b_1^{'}}{2}+b_1K \big ) \Big \{ \big (\frac{(1-\sigma )b_1^{'}}{2}+b_1K \big ) \big (K_1 \gamma _1^2-e \big ) \Big \}$.